Maths is a web, not a list
Topics that seem separate (fractions, geometry, algebra, probability) are deeply connected. Seeing those connections makes maths easier to learn and remember, and lets you solve problems in more than one way.
Example 1: The same idea in many forms
Proportional reasoning appears everywhere:
| Topic | Form |
|---|---|
| Fractions | equivalent fractions: 2/3 = 8/12 |
| Ratios | 2:3 = 8:12 |
| Percents | 40% = 40/100 |
| Algebra | y = kx |
| Geometry | similar triangles |
| Measurement | scale drawings |
| Science | speed = distance/time |
Example 2: Algebra meets geometry
- The equation y = 2x + 1 is a line on a graph.
- (a + b)² = a² + 2ab + b² is an area of a square split into four pieces.
- The Pythagorean theorem links algebra (a² + b² = c²) and geometry (squares on triangle sides) and leads to the distance formula.
Example 3: Number patterns and shapes
Square numbers (1, 4, 9, 16) make squares of dots. Triangular numbers (1, 3, 6, 10) make triangles. Adding two consecutive triangular numbers gives a square number: 3 + 6 = 9.
Example 4: Probability and fractions
P(rolling an even number) = 3/6 = 1/2 = 0.5 = 50%. Probability is fractions applied to chance.
Example 5: Functions everywhere
| Situation | Function type |
|---|---|
| Earnings at a fixed hourly rate | linear |
| Area of a square vs side | quadratic |
| Bacteria doubling | exponential |
| Tides, sound waves | trigonometric (sinusoidal) |
Multiple representations
A single relationship can be shown as words, a table, a graph and an equation. Each representation reveals something different. Moving between them is one of the most powerful connections in maths.
Connections beyond maths
Maths is interdisciplinary:
- Science: formulas, data analysis, rates
- Art and design: symmetry, perspective, the golden ratio
- Music: rhythms as fractions, frequencies as ratios
- Geography: map scales, coordinates, population growth
- Computer science: binary, logic, algorithms
- Economics: percent change, interest, graphs of supply and demand
Worked examples
Example 1: Show 3/4 in four representations. 0.75, 75%, 3:4 (part-to-whole), a 4-part bar with 3 shaded.
Example 2: Solve x² = 49 using geometry. A square with area 49 has side 7.
Example 3: Connect slope and rate. The slope of a distance–time graph is the speed.
Common mistakes
- Learning each topic in isolation.
- Sticking to one method when another representation makes the problem easier.
Try it yourself
- Write 1/5 as a decimal, a percent and a ratio.
- Which geometry idea explains (x + 3)² = x² + 6x + 9?
- Give an example of maths used in music.
- What does the slope of a cost-vs-items graph represent?
Answers: 1) 0.2, 20%, 1:5 (part-to-whole) 2) the area of a square split into four pieces 3) e.g. note lengths as fractions 4) the price per item
Quick recap
- Maths topics are linked by big ideas like proportional reasoning and functions.
- Use several representations of the same idea.
- Maths connections extend into science, art, music and more.